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For every n ≥ 2 we construct an n-dimensional compact subset X of some Euclidean space E so that none of the canonical projections of E on its two-dimensional coordinate subspaces has a stable value when restricted to X. This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of X and in particular its sublattice that consists of monotone decompositions when X is hereditarily indecomposable

Stability and dimension—a counterexample to a conjecture of Chogoshvili